There is one former Twin on the BBWAA's Hall of Fame ballot this year: newcomer Luis Castillo.
Castillo will be a one-and-done guy on the ballot, and that's as it should be, even if the most similar player to him in Baseball Reference's "similarity score" is enshrinee Johnny Evers. Evers was a debatable choice himself, and nobody else on the list is in, But Castillo was a fine player for a few seasons: three-time Gold Glove winner, two-time All-Star, regular on the 2003 World Series winning Marlins, two-time NL stolen base leader.
Castillo played second base for Minnesota in 2006-07. The Twins picked up the second baseman from the Marlins for a pair of minor league pitchers (Scott Tyler and Travis Bowyer) who reached the majors but didn't do much, and he was a significant part of the "pirahanas" team that had such a glorious season in 2006.
When the Twins fell out of contention in 2007, Terry Ryan -- in what I beleive was the last significant move he made in his first term as general manager -- traded Castillo at the deadline to the Mets for a pair of minor leaguers. Outfielder Dustin Martin never reached the majors, but catcher Drew Butera did. Butera spent part or all of four seasons with the Twins, and then they traded him (another deadline deal) in 2013 to the Dodgers for minor league lefty Miguel Sulbaran.
Sulbaran finished 2013 in low A for the Twins. The next April, the Twins traded him to the Yankees for Eduardo Nunez, who remains on the big league roster as a bench piece. So a decade after the Twins traded for Castillo, the move still has a presence.